Zhe  •Qlnlversiti?  of  Cblcaflo 

FOUNDKD  BY  JOHN  D.  ROCKEFELLER 


THE  TERNARY  LINEAR  TRANSFORMATION 
GROUP  G3.360  AND  ITS  COMPLETE  INVA- 
RIANT SYSTEM 


A  DISSERTATIO 


SUBMITTED    TO    THE    FACULTIES    OF    THE    GRADUATE    SCHOOLS    OF   ARTS, 

LITERATURE,  AND    SCIENCE,  IN    CANDIDACY    FOR    THE 

DEGREE    OF    DOCTOR    OF    PHILOSOPHY 


(department  of  mathematics) 


BY 

GEORGE    LINCOLN    BROWN 


PRINTED    BY 

Jlbc  "GimvecsttB  of  dbicago  press 

CHICAGO 


ITbe  XflnlvergltB  of  Chicago 

FOUNDED  BV  JOHN  D.  ROCKEFELLBR 


THE  TERNARY  LINEAR  TRANSFORMATION 
GROUP  G3.360  AND  ITS  COMPLETE  INVA- 
RIANT SYSTEM 


A  DISSERTATION 

SUBMITTED    TO    THE    FACULTIES    OF    THE    GRADUATE    SCHOOLS    OF   ARTS, 

LITERATURE,  AND    SCIENCE,  IN    CANDIDACY    FOR    THE 

DEGREE    OF    DOCTOR    OF    PHILOSOPHY 

(department  of  mathematics) 


BY 

GEORGE   LINCOLN    BROWN 


PRINTED    BY 

{Tbc  Tantvergitg  ot  Cbicago  press 

CHICAGO 


EXCHANGE 


.     AND  ITS  COMPLETE  INVARIANT  SYSTEM.' 

SECTION    I.       INTRODUCTION. 

The  problem  of  verifying  the  existence  of  the  coUineation  group 
G360  in  a  definitive  manner  and  of  finding  its  complete  invariant  system 
was  proposed  to  me  by  Professor  Moore,  of  the  University  of  Chicago, 
his  suggestion  being  prompted  by  H.  Valentiner's  deterniinaticn^cf  this 
group.*  While  the  latter  gives  the  method. by  which  the  generators  >if 
the  group  may  be  found,  he  does  not  compute  the-r  delin'ltivV  forms/'  ' 

After  having  deduced  the  complete  invariant  system  under  the 
direction  of  Professor  Maschke,  the  solution  of  the  problem  was  given 
by  me  before  the  Mathematical  Club  of  the  university,  August  21, 
1896.  Soon  after,  my  attention  was  called  to  an  article  by  A.  Wiman,' 
which  had  just  been  received  at  the  university,  and  which  is  a  discus- 
sion of  this  same  group.  I  have  thought  it  proper,  therefore,  after 
giving  my  own  discussion,  to  indicate  very  briefly  the  interesting 
method  which  he  employs  to  determine  the  complete  invariant  system, 
at  the  same  time  showing  that  it  is  equivalent  to  that  found  by  myself. 

SECTION    II.       THE    GENERATORS    OF    THE    GROUP    G3.360. 

Since  the  cube  and  the  fifth  roots  of  unity  enter  into  the  coefficients 
of  the  operators  of  our  group,  for  convenience  in  calculation  we  make 
use  of  the  fifteenth  roots  of  unity, 

(«  =  o,  I,  2,  .  .   .  14) 

For  these  we  have  the  reducing  equation  : 
derived  from  (^.5_i)(^_,)  ^ 

1  For  literature  on  this  subject  see:  H.  Valentiner,  De  endelige  Transformations- Cruppers 
Theori,  pp.  192-8;  A.  Wiman,  "  Ueber  eine  einfache  Gruppe  von  360  ebenen  Collineationen,"  Math. 
Annalen,  Bd.  XLVII,  pp.  S31-56;  F.  Gerbaldi,  "  Sul  Gruppo  semplice  di  360  coUineazione  pione," 
Palermo  Rend.,  T.  XII,  pp.  23-94;  T.  XIII,  pp.  161-99;  R.  Fricke,  "Ueber  eine  einfache  Gruppe 
von  360  Operationen,"  Gott.  Nachr.,  1896,  pp.  199-206;  L.  Lachtin,  "Die  Differentialresolvente  einer 
algebraischen  Gleichung  6ten  Grades  mit  einer  Gruppe  36oster  Ordnung,"  Math.  Annalen.,  Bd.  LI,  pp. 
463-72. 

2  De  endelige  Transformations-Gruppers  Theori,  pp.  192-8. 

3"  Ueber  eine  einfache  Gruppe  von  360  ebenen  Collineationen,"  Math.  Annalen,  Bd.  XLVII,  pp. 
531-56  (issued  August  16,  1896). 

3 


'sc^^sei 


4  THE  TERNARY  LINEAR  TRANSFORMATION  GROUP  03.360 

by  means  of  which  any  one  of  them  can  be  expressed  as  a  linear  func- 
tion of  the  eight  roots, 

(«  =  O,   I,  2,  .   .   .  7) 

For  the  generators*  of  the  collineation  group  G360  we  have  com- 
puted the  definitive  forms : 

fji.x'  =  X    , 

U:    fxy'  =  e^y  , 

vx  ^-  ^—7^-^r  +  ' y  +  z  . 

V:      vy'= -=^x  +        ^       y  J^ 

vz'=     ^  ' X  +  ^  H 7^--2   , 

2iri 

i/5  =  93(i_e3)  (i_e6)  . 

6^ and  Fare  of  periods  5  and  2,  respectively,  and  for  each  of  them 
the  determinant  of  the  coefficients  is  unity.  Since  we  shall  deal  with 
the  substitution  group  instead  of  the  group  of  collineations,  we  choose 
^3^  v3  =  I,  i.  e.,\i.,v  =  i,  6^  or  ^'°.  If  ii=6^  or  6^°,  it  is  evident  that 
i7  as  a  substitution  has  the  period  15,  while  for  the  same  values  of  v, 
V  has  the  period  6 ;  in  either  case  the  cyclic  self-conjugate  subgroup 
generated  by 

x'=e^x, 

y'  =  e^y  , 

z'=e^z, 

which  groups  we  name  G3,  will  belong  to  our  group,  and  consequently 
each  operator  of  the  collineation  group  will  appear  at  least  three  times 
in  the  substitution  group,  with  the  multipliers  i,  ^^  and  6^°.  For  the 
remaining  case,  /u.=:v  =  i,  £/"  and  V  as,  substitutions  have  the  same 
periods  as  when  collineations.    We  shall  find,  however,  that  G3  appears 

1  See  Valentinhr,  De  endelige  Transforntations-Gruppers  Theort,  p.  192.  The  notation  of 
the  operators  has  been  changed  from  A  and  B  to  U  and  F  respectively. 

2  The  notation  G«  will  be  used  throughout  this  paper  to  indicate  in  general  a  group  of  order  «, 


THE  TERNARY  LINEAR  TRANSFORMATION  GROUP  03.360  5 

in  the  group  in  this  case  also  ; '  and  accordingly  we  shall  take  here- 
after, without  further  specification,  ^^v  =  1. 

We  notice  that  the  coefficients  of  V  are  all  real,  and  that,  if  the 
complex  conjugates  of  the  coefficients  of  U  are  taken,  U*  is  obtained. 
From  this  can  be  concluded  : 

1.  The  complex  conjugates  of  the  operators  of  our  group  also 
belong  to  the  group. 

2.  The  complex  conjugates  of  the  operators  of  any  subgroup  of 
our  group  also  form  a  subgroup. 

3.  If  any  expression  is  invariant  under  the  whole  group,  its  com- 
plex conjugate  is  invariant,  and  consequently  its  real  and  imaginary 
parts  are  both  invariant ;  hence  we  may  choose  the  invariant  system 
real. 

SECTION    III.      THE    ICOSAHEDRON    SUBGROUPS    OF    G3.360. 

The  order  of  G3.360  is  determined  by  means  of  these  subgroups. 
G3.360  is  found  to  be  isomorphic  with  the  group  of  even  permutations  of  six 
things. 

We  combine  t/'and  Fby  the  general  method  for  multiplying  trans- 
formations and  form 

e^X    =        -— y    -\^    —^z    , 


1/5  1/5  VS     ' 

2 

2 

The  second  power  of  this  operator  is 

x'=^e^x  , 
{U^VUVU^Vy.-    y'=d^y  , 

whence  we  see  that  our  group  contains  G3 ,  and  consequently  the  same 
operators  as  when  /u.  or  1/  is  taken  equal  to  6^  or  6^°. 

If  in  the  generators  of  the  ternary  icosahedron  group  as  given  by 
Professor  Klein,^ 

I  See  sec.  3.  a  Klein,  VorUsungen  iiber  das  Ikosaeder,  pp.  213-ig, 


6  THE  TERNARY  LINEAR  TRANSFORMATION  GROUP  03.360 

Ao  ==  Ag  , 

s.-    a;=€A,  , 

A^  =^  fr  A^  , 

VIa:=a,+a^^a,  , 

27r» 

£  =  *  T   =  fl3 

the  substitution 

Ao=^x  , 

A,=  -  iS^ky  , 

A,=i6^^kz  , 

be  made,  the  operators  £/"  and  {U^  VUVU^  Vy  of  our  group  are 
obtained.  The  group  G^  is  consequently  a  subgroup  of  our  group. 
The  invariant  system  of  the  G^o  just  mentioned  is  obtained  by  making 
the  above  substitution  in  the  expressions'  which  are  given  by  Professor 
Klein,  and  are  : 

A^  =  3^ -{- kf" yz  , 

B,=  8x*yz-2k'x'y'z'+k*y^z^-i-tk^x{y^- z^)  , 

Q=^20X^y'z'—  i6oA'x*y^z^-{-  2oJ^x'y*z*-{-  6k^y^z^-\- 

4ik  (y^  —  25)  (32  x*-  20k' x'yz-\-  s ^y' z')  -  k"" (7'°  +  2")  , 

D,  =  U(B^-y  +  ^""0)  n  1(2  e^+  2  ^')  ^  +  le^-ky  -  iO"- kz\ 

n\{2e^-\-2d'')x  +  ie^^ky -i6"^kz\  . 

V 

Corresponding  to  this  subgroup  is  the  complex  conjugate  subgroup 
which  has  as  an  invariant  system  the  complex  conjugates  of  these 
expressions,  for  which  we  shall  use  the  notation,  A^,  B^,  C„  Z>,. 

By  successive  applications  of  U  and  V,  A^  is  found  to  take  in  all 
eighteen  expressions,  six  expressions  which  we  shall  call  Ay  (v  =  i,  2, , . ,  6), 
and  each  of  these  multiplied  by  6^  and  6^°. 

The  manner  in  which  these  quadratic  expressions  are  permuted  by 
6^  and  Fcan  be  seen  from  the  following  table,  in  which  U  and  V  act 
upon  the  quadratic  forms  which  are  written  in  the  upper  horizontal 
line : 

I  Ibid.,  pp.  217,  218. 


THE  TERNARY  LINEAR  TRANSFORMATION  GROUP  €3.360 


A, 

A, 

A, 

A, 

A, 

Ae 

u 

A, 

A, 

A, 

As 

A, 

A, 

V 

A. 

A, 

O^A, 

e-A, 

A, 

Ae 

the  six  quadratic  expressions  being 
A,  =  x'-\-k'yz  , 


(v  =  2,  3,.  .  .6)     >> 


The  complex  conjugates  of  these  quadratic  expressions,  A,,  (v  =  1, 2, . . .  6) , 
are  permuted  in  exactly  the  same  way  by  U'*  and  F. 

Now,  every  ternary  transformation  group  for  whose  operators  the 
determinants  of  the  coefficients  are  unity,  and  which  leaves  a  quadratic 
form  unchanged,  is  holoedrically  isomorphic  with  a  binary  collineation 
group,'  and  since  there  is  no  binary  collineation  group  of  order  6o« 
except  for  n  —  1,''  the  operators  which  leave  any  one  of  the  quadratic 
expressions  Ay  unchanged  must  all  be  contained  in  the  Ggo  correspond- 
ing. Hence,  by  the  general  group  theory,  the  order  of  the  group 
generated  by  U  and  F  is  60  X  18  =  1080. 

If  U  and  Fare  written  as  permutations  of  the  six  conies  gotten  by 
placing  yi>'  (v  =  i,  2, .  .  .  6)  equal  to  zero,  we  have 

U={A^A,A,A,A,), 
F={A,A:){A,A,). 

These  generate  the  group  of  even  permutations  of  six  things.  The  Geo 
to  which  Aj  belongs  is  generated  by 

C/={A,A^A,A,A,), 
{U^  VUVU^  Vf  =  {A^A,){A,A,)  , 

which  are  generators  of  the  group  of  even  permutations  of  five  things. 
It  is  evident,  then,  that  the  permutations  of  the  five  conies  A^  {y=2, 3, ...  6) 
under  the  Ggo  to  which  A^  belongs  form  the  group  of  even  permuta- 

1  Weber,  Lehrbuch  der  Algebra,  Bd.  II,  Abschnitt  vii,  sec.  49,  pp.  191,  192. 
'Ibid.,  sec.  52,  p.  203. 


8  THE  TERNARY  LINEAR  TRANSFORMATION  GROUP  03.360 

tions  of  five  things  with  which  group  the  Ggo  is  holoedrically  iso- 
morphic. And  since  the  six  conjugate  G^o  are  simple,  they  have  no 
operator  in  common  other  than  the  identity,  and,  therefore,  by  the 
general  theory  of  isomorphic  correspondence,'  the  collineation  group  of 
U  and  Fis  holoedrically  isomorphic  with  the  group  of  even  permuta- 
tions of  six  things. 

We  have  found  that,  owing  to  the  presence  of  G3 ,  every  operator  of 
the  collineation  group  appears  at  least  three  times  in  the  substitution 
group,  with  the  multipliers  i,  6^,  and  6^°.  Since  the  order  of  the  sub- 
stitution group  is  1080,  each  operator  of  the  collineation  group  corre- 
sponds to  only  three  operators  in  the  substitution  group.  It  is  evident 
that  this  correspondence  determines  the  meriedric  isomorphism  of  the 
substitution  group  with  the  collineation  group  of  U  and  V,  to  the 
identity  of  the  latter  group  corresponding  the  group  G3  of  the  former. 
To  indicate  this  isomorphism  we  use  the  notation  G3.360. 

The  interesting,  but  not  surprising,  analytical  relation  existing 
between  the  two  systems  of  quadratic  forms  belonging  to  the  Ggo  sub- 
groups can  be  seen  from  the  identity 

{k'- 2)  A,  =  A, -^6'°^ Ay   . 

(f  =  2.  3,  ...  6) 

SECTION    IV.       THE    INVARIANT    SYSTEM    OF    G3.360  DETERMINED    BY  MEANS 
OF    THE    Gfio    SUBGROUPS. 

From  the  nature  of  our  group  (since  it  contains  G3')  it  is  seen  that 
every  invariant  under  it  must  be  of  degree  3«  («  =  -f-  integer).  Also, 
since  it  contains  Geo  as  a  subgroup,  its  invariants  must  be  integral 
functions  of  the  expressions  which  form  the  invariant  system  of  the 
Gfio  subgroup.  The  invariants  of  G3.360  are  consequently  integral  func- 
tions of  A^,  B^,  Q,  and  Z>,  (see  sec.  3).  Now,  instead  of  B^  and  C,, 
we  can  take  as  invariants  of  Ggo 

(V=I,2,    ...    6) 

and 

C;  =  UAy  ; 

(I'-a.a,  ...  6) 

for  from  the  way  in  which  the  quadratic  forms  are  permuted  under  the 
group  G3.360  it  is  evident  that  B^  and  C  are  invariant  under  the  Ggo, 
which  leaves  A^  unchanged ;    moreover,  it  is  easily  determined  that 

I  See  BOLZA,  Om  the  Theory  of  Substitution  Groups  and  its  Application  to  Algebraic  Equa- 
tions, sec.  8,  art.  39. 


THE  TERNARY  LINEAR  TRANSFORMATION  GROUP  03.360  9 

A^,  B[  ,  and  C/  are  independent,  since  B^  is  not  the  cube  of  A^,  and 
C,'  contains  terms  (for  instance,  in  y^°,  2'°)  which  cannot  be  gotten  by 
combining  A^  and  -5/  in  an  integral  function.  But  B^  and  A^  C/ 
are  both  invariant  under  G3.360  and  independent  of  each  other.  We 
use  for  them,  therefore,  the  notation  T^g  and  i^',,,  respectively. 

Every  invariant  of  the  sixth  degree  under  G3.360  must  have  the  form 
aAl-\-fiFe  , 

where  a  and  /3  are  constants.  A^  not  being  invariant  under  the  main 
group,  ^6  must  be  the  only  invariant  of  this  degree. 

Likewise,  every  invariant  of  the  twelfth  degree  can  be  written 
aAl-]-(3AlJ^,+  yFl  +  8A,C\  . 

Now,  since  F^  and  A^  C^  =  F^^  are  invariant,  any  invariant  of  the  twelfth 
degree  independent  of  these  must  have  the  form 

aA\  +  pA\F,   • 

but,  since  A\  is  a  divisor  of  this  invariant  form  (if  any  such  exist),  and 
A^  under  G3.360  is  permuted  with  each  of  the  other  quadratic  forms 
Ay  {y  =  i,  ■>,,..  .  6),  the  cubes  of  each  of  these  other  five  quadratic  forms 
must  be  divisors  of  the  expression,  which  is  impossible. 

Similarly,  it  can  be  shown  that  there  are  no  invariants  of  the  eight- 
eenth or  twenty-fourth  degree  independent  of  Ff,  and  F[,. 

Again,  it  is  evident  that  a  new  invariant  independent  of  F^  and 
F[^  is  given  by 

>'  =  1,  2,  .  .  .  6 

since  this  contains  terms  (for  instance,  in  y°  and  z^)  which  cannot  be 
gotten  by  combining  F^  and  F^^ . 

By  applying  the  operator  Voi  our  group  to 

aAl-\-pB,  =  \F, 

and  equating  the  coefficients  of  corresponding  terms  in  the  original 
and  the  transformed  expressions,  the  ratio 


/8       15 -3^^^ 

is  gotten,  which  yields,  on  removing  the  constant  factor  \=  8, 

Fi  =  x^-\- 15  x*yz—  isx'y'z'-  loy^z^—  ^Vixiy^—z^)  . 

The  invariant  F^^  has  been  given  as  the  product  of  the  six  quad- 
ratic forms  Ay  (v  =  i,  s,  .  .  .  6).     It  can  easily  be  verified  by  an  inspection 


10         THE  TERNARY  LINEAR  TRANSFORMATION  GROUP  03.360 

of  the  terms  that  form  the  invariant  of  the  twelfth  degree,  that  the  Hes- 
sian covariant  of  F^,  which  we  shall  call  F^,,  may  be  taken,  and  for 
the  invariant  of  the  thirtieth  degree,  the  same  function  of  F^,,  which 
we  name  -^30 .     We  then  have  these  invariants  real. 

The  F^^  is  found  to  be 
F^^  ==  ar"  —  1 8  x^^y  z  +  1 5  o^f  2°  +  40  x^y^  2^+345  x^f  ^  + 

156  oc'y'^  z^  —  i%f  z*  —  V I X  (^s  _  gS)  [36  ;x:*  +  1 2  x^y  z  + 

1 20  oc'y'^—  3o>'3  23]  —  3  (/°  +  2")  (^4-2^2)  . 

On  account  of  its  length,  we  have  not  determined  the  expression 
fori^3o. 

We  have  proved  the  following  relations  : 

(i)    ^Al=f^{F^=%F,, 

V 

(2)  UA^=/,{F,,  i^„)=^;. , 

V 

(3)  ^Al=MF,,  F^,), 

(4)  ^Al=f,{F,,  7^„)  , 

(5)  ^A-=f,{F,,  ir„)  , 

V  \ 

(6)  ^A^^=f,{F,,  i^„,  f^)  =  f;,  , 


where//  (« =  i,  2,  ...  6)  are  integral  functions  of  i^g,  i^.^,  and  -^30.  Con- 
sequently, every  symmetric  function  of  the  cubes  of  the  six  quadratic 
forms  Ay  (v  =  i,  2,  .  .  .  6)  is  an  integral  function  of  F(„  F„,  and  F^,. 

We  can  now  prove  that  every  invariant  of  even  degree  of  G3.360  is 
an  integral  function  of  F^,  F^^,  and  F^.  For,  let  I^„  be  any  such 
invariant,  then 


(l)        /.n=    2     S     2    ^P-^^^^L^J 


p,  c,  T=o,  I,  2,  3  .  .  .  X^  X^  X^  [3      pp  F''  A"-' 

6p  +  2<r  +  ior=2«  2^   2^   2^Pp<TT^^-f^^^^        • 

Applying  the  operators  of  our  group  to  this  form  so  as  to  make  A^ 
assume  successively  the  eighteen  forms  B^^  A^  (Il  =  oii',2' '  ^)'  ^"^^ 
adding,  we  obtain 


THE  TERNARY  LINEAR  TRANSFORMATION  GROUP  03.360        1 1 

(.)  /.,=  2  2  2  ^'"^'^"  2  2 f""'^'*""'  ■ 

Those  terms  for  which  o-  —  t  4=  o  (mod.  3)  drop  out. 
For  0-  —  T=:3z«  (m  =  -\-  integer), 

(3)  22^^''  ^•'^''"' = 3  2 ^'"'  ^  ^  ^'^"""^  ^^' '  ^" '  ^-^ ' 

where  <t>^  is  an  integral  function  of  J^6>  -^12.  and  -^^30  of  degree  2  (<r  —  t). 
For  a  —  T=^  —  ^m(m=-j-  integer), 

<f>,  being  the  sum  of  the  products  to  the  power  t  —  o-  of  the  six  quad- 
ratic expressions  in  sets  of  five,  and  consequently  an  integral  function 
of  J^6,  -^12  >  and  7^30  of  degree  10  (t  —  a).  Substituting  the  values  of 
the  expressions  (3)  and  (4)  in  (2)  we  get 

(5)  ^-=3222  ^"'^  ^'  ^"  *^^^"'^  ^^' '  ^" '  ^-^ + 

p  O-  T 

<r  —  T  =  3  wj  (wz  =  0,  I,  2,  3  .  .  .) 

3  2  2  2  yp-^^^^'<^^°^^-'^(^-^-^3o), 

p  <r  T 

p  —  T  =  —  3  »«   (»»  =  1,    2,    3    .    .    .) 

/.  <f.,  /^„  is  an  integral  function  of  J^^,  -^12  >  and  -F30,  since  i^^  is  an 
integral  function  of  J^e  and  ^,2 . 

Since  D^  is  the  only  fundamental  invariant  of  odd  degree  under 
the  Gfio  to  which  it  belongs,  it  must  be  a  divisor  of  every  invariant  of 
odd  degree  under  G3.360.  The  same  is  evidently  true  of  ^^.  There 
cannot  be,  consequently,  more  than  one  fundamental  invariant  of  odd 
degree ;  and  if  one  such  invariant  exists,  it  consists  of  the  fifteen  linear 
factors  of  Z>j,  together  with  the  factors  conjugate  to  these  under  G3.360. 
Now,  since  Z),  remains  invariant  under  a  Geo  and  also  under  G3',  or 
under  a  group  G3.60,  it  cannot  take  more  than  six  forms  under  G3.360. 
If  we  multiply  together  these  six  conjugate  expressions,  an  expression 
of  the  ninetieth  degree  in  x,  y,  z  is  obtained,  which  we  call  <^9o-  In 
<t>go  each  linear  factor  occurs  as  often  as  every  other  factor,  and  con- 
sequently ^go  is  the  first,  second,  third,  fifth,  or  sixth  power  of  an 
invariant  of  G3.360 ,  according  to  the  number  of  times  each  factor  occurs. 
If  each  factor  occurs  an  odd  number  of  times,  ^g,,  is  some  power  of  an 


12         THE  TERNARY  LINEAR  TRANSFORMATION  GROUP  03.360 

invariant  of  even  degree,  and  there  is  no  invariant  of  odd  degree.  If 
each  factor  occurs  six  times,  Z>,  itself  is  invariant  under  G3.360,  and  also 
Z>i ,  which  is  impossible,  since  D^  is  different  from  D^ .  Consequently, 
if  there  is  any  fundamental  invariant  of  odd  degree,  it  must  consist  of 
forty-five  linear  factors,  each  of  which  occurs  twice  in  <^g^. 

Now,  we  have  an  invariant  of  odd  degree  in  the  functional  deter- 
minant of  F^,  i^j2,  and  F^„,  which  can  easily  be  proved  not  identically 
equal  to  zero,  and  which  we  name 


dx 

'5 
dx 

^45- 

9^« 
dy 

3^. 

hy 

dy 

8^6 

8. 

Bz 

In  Z>,  and  its  conjugate  Z>,  we  have  twenty-five  of  the  linear  factors 
of  i^45 ,  and,  by  applying  the  operators  of  our  group,  the  remaining 
twenty  factors  can  easily  be  found. 

It  is  evident  that  F^^,  being  of  even  degree,  is  an  integral  function 
of  Fq,  i^i2,  and  F^^.  On  account  of  the  complexity  of  the  expressions 
involved,  we  have  not  computed  this  analytical  relation.  We  may, 
however,  express  F^^  in  a  very  simple  manner  through  the  six  quadratic 
forms  which  represent  the  Ggo  conies  of  one  system.  For  the  points  of 
intersection  of  these  conies  with  each  other  lie  on  the  lines  of 


A.- A, 


gives  us  two  of  these  lines.  By  applying  the  operators  [/  and  F  to 
this  difference,  we  can  obtain  the  entire  number  of  lines.  This  differ- 
ence takes  forty-five  forms  under  the  group  (neglecting  the  factors  $^ 
and  0'°),  viz., 

'i  <  k;  i,k  =  z,i,  .  ,  .6\ 
v  =  o,  I,  2         / 


('■ 


and,  consequently,  each  linear  factor  is  repeated,  and  we  have 
F:,  =  TL{A,-6^^A,). 

V,  i,  k 


THE  TERNARY  LINEAR  TRANSFORMATION  GROUP  63.360        13 

SECTION    V.       THE     INVARIANT     SYSTEM    DETERMINED    BY    MR.    WIMAN    IS 
IDENTIFIED    WITH    THAT    OF    SECTION    IV. CONCLUDING    REMARKS. 

Since  the  operators  of  the  coUineation  group  G360  have  been  related 
to  those  of  the  permutation  group  G360  by  means  of  the  permutations 
which  C/'and  F  effect  on  the  six  Geo  conies  of  one  system,  the  subgroups 
of  the  coUineation  group,  and  consequently  those  of  the  substitution 
group  G3.360,  can  be  determined  through  the  subgroups  of  this  permu- 
tation group.  We  have  seen  that  F^,  F^^,  F^^,  and  F^^  can  be  expressed 
as  integral  symmetric  functions  of  the  six  quadratic  expressions  which 
represent  these  conies.  For  the  many  interesting  features  of  the  sub- 
groups of  the  coUineation  group  G360  and  their  configurations,  the 
reader  is  referred  to  the  authors  cited  in  sec.  i  of  this  paper.  In 
concluding,  it  is  merely  our  purpose  to  show  briefly  how  Mr.  Wiman 
determines  the  invariant  system  of  the  group  G3.360,  and  that  it  is  equiva- 
lent to  that  of  sec.  4. 

The  group  G3.360  is  reached  by  Mr.  Wiman  through  the  G^^  sub- 
groups. In  the  coUineation  group  G360  there  are  two  systems  each  of 
fifteen  conjugate  G^^,  and  corresponding  to  these  two  systems  each  of 
six  conjugate  Ggo  subgroups  (see  sec.  3),  two  Geo  oi  the  same  system 
having  a  Gi^  in  common,  which  belongs  to  a  G^  under  the  whole 
group. 

In  the  coUineation  group  G360  are  forty-five  operators  of  period  2, 
whose  perspective  axes  form  the  two  systems  of  self-conjugate  triangles 
belonging  to  the  G^^.  These  axes  are  evidently  the  lines  given  by 
(sec.  4) 

F,,  =  o  . 

Each  of  the  linear  forms  representing  these  axes  is  found  to  occur 
three  times,  with  the  cube  roots  of  unity  as  factors.  The  presence  of 
these  roots  indicates  that  the  subgroup  G3'  (sec.  2)  appears  in  the  sub- 
stitution group. 

The  normal  form,  of  the  invariant  of  the  sixth  degree  is  given  by 
Mr.  Wiman  as 

Ce=  10:^373+  ^z{x^-\-y^)—  4S  x'y^z'—  is5xyz'-\-  2Tz\ 
and  two  Geo  conies  which  belong  to  different  systems  are 

The  normal  form  of  the  Ce  above  is  transformed  into  F(,  of  sec.  4  by 
the  substitutions 


14         THE  TERNARY  LINEAR  TRANSFORMATION  GROUP  €3.360 


y  =-y  , 

X 

which  also  transforms  the  two  conies  corresponding  into  A^  and  its 
conjugate  ^^  (see  sec.  3). 

After  having  made  a  study  of  the  subgroup  configurations  of  the 
collineation  group  G350 ,  Mr.  Wiman  affirms  that  on  the  curve 

F,  =  o 

are  only  three  special  point  systems.  These  consist  of  the  72  points 
of  tangency  of  the  Geo  conies  of  one  system  with  those  of  the  other 
system,  and  the  270  points  in  which 

F,  =  o 

is  cut  by  the  forty-five  perspective  axes,  these  points  being  divided  into 
two  closed  systems  of  90  and  180  points,  respectively. 

Since  the  invariant  of  the  sixth  degree  found  by  Mr.  Wiman  has 
been  shown  to  be  equivalent  to  that  of  see.  4,  we  may  without  con- 
fusion use  for  it  here  the  notation  Ff, .  The  Hessian  H  of  F(,  is  given 
as  the  invariant  of  the  twelfth  degree,  and  the  curve 

is  found  to  cut  out  on 

F,  =  o 

the  special  system  of  72  points. 

An  invariant  of  the  thirtieth  degree  independent  of  F^,  and  H  is 
given  as 


dy 
hH 


dyhx 

dzhx 

^F, 

^^F, 

h^F, 

^  = 

dxdz 

dydz 

Szdy 

B'F, 
80^ 

hH 

hH 

dif 

dx 

^ 

dz 

the  curve 

^  =  0 

cutting  out  on 

F, 


THE  TERNARY  LINEAR  TRANSFORMATION  GROUP  G^.^bo        1 5 

the  special  system  of  90  points  doubly  counted,  the  two  curves  having 
simple  contact  in  each  of  these  points. 

The  functional  determinant  of  Ff,,  H,  and  $  is  found  to  be  an 
invariant  of  the  forty-fifth  degree  ^,  and,  as  has  been  stated,  represents 
the  forty-five  perspective  axes  of  the  collineation  group  G^. 

By  means  of  the  point  systems  on 

F,  =  o 

it  is  now  proved  that  every  invariant  of  the  group  can  be  expressed 
through  F^,  H,  ^,  and  *;  also,  that  *'  is  an  integral  function  of  F^, 
H,  and  $. 

We  have  identified  the  Ff,  given  by  Mr.  Wiman  with  that  of  sec. 
4  by  a  very  simple  transformation.  From  the  subsequent  steps  in  the 
two  methods  it  is  evident  that  the  two  invariant  systems  are  equivalent. 


VITA. 

George  Lincoln  Brown  was  born  January  25,  1869,  in  Bates 
county,  Missouri.  He  entered  the  Preparatory  Department  of  the 
University  of  the  State  of  Missouri  in  September,  1885,  and  attended 
that  institution  irregularly  during  the  next  few  years,  receiving  the 
degree  of  Bachelor  of  Science  in  June,  1892.  He  returned  to  the 
Missouri  University  as  Teaching  Fellow  in  Mathematics,  and  during 
the  next  two  years  pursued  advanced  work  in  this  department  under 
the  direction  of  Professors  W.  B.  Smith  and  W.  C.  Tindall,  receiving 
the  degree  of  Master  of  Science  in  1893. 

During  the  years  1894-95  and  1895-96  he  attended  the  University  of 
Chicago  as  Fellow  in  Mathematics,  and  pursued  graduate  courses  in 
this  department  with  Professors  E.  H.  Moore,  O.  Bolza,  and  H. 
Maschke,  and  with  Dr.  Kurt  Laves  in  Mathematical  Astronomy. 

Mr.  Brown  desires  to  express  here  his  gratitude  to  the  instructors 
mentioned  above  for  their  direction  and  words  of  encouragement,  and 
to  thank  them  especially  for  the  inspiration  which  their  lives  have 
afforded  him. 


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